Study Notes · Part 1: Fundamental Concepts & OCV

Date: 2026-05-21
Task: Build an electrochemical battery simulation model from scratch in Python
Progress: SPM Single Particle Model · OCV Curves Complete

1. What Can Electrochemical Simulation Do?

Electrochemical simulation = creating a "digital twin" of a battery — using mathematical equations to simulate everything happening inside a battery on a computer.

2. What Happens Inside a Li-ion Battery?

Three key steps during discharge:

graph LR
    A["Anode Particle<br/>Li → Li⁺ + e⁻"] -->|"Li⁺ through electrolyte"| B["Cathode Particle<br/>Li⁺ + e⁻ → Li"]
    A -->|"e⁻ through external circuit"| C["Work!<br/>Light the bulb"]
    C --> B

1. Deintercalation: Lithium atoms at the anode particle surface lose electrons, becoming Li⁺ that enter the electrolyte
2. Migration: Li⁺ travels through the electrolyte (separator) toward the cathode
3. Intercalation: Li⁺ gains electrons at the cathode particle surface and embeds into the cathode

3. What is SPM (Single Particle Model)?

SPM = Single Particle Model, the simplest electrochemical battery model.

graph TD
    subgraph Real Battery
        A1["Particle Population<br/>Thousands of particles"]
        A2["Particle Population<br/>Thousands of particles"]
    end
    subgraph SPM Model
        B1["One sphere = Entire anode"]
        B2["One sphere = Entire cathode"]
    end
    A1 -.->|"Simplification"| B1
    A2 -.->|"Simplification"| B2

Three Core Assumptions of SPM

SPM Capability Boundaries

• Voltage–time curves (discharge/charge curves)
• Lithium concentration distribution within particles
• Battery performance under different currents
• SOC evolution over time
• Accurate prediction at high currents (neglects electrolyte gradients)
• Differences between particles at different positions

4. Parameter Naming Convention

Naming formula: self.{quantity}_{phase}_{position}

Quantity Prefixes

Phase Suffixes & Position Suffixes

Note: _s can denote either "solid phase" or "separator" depending on context. For example, eps_s_n = negative solid-phase volume fraction, eps_e_s = separator electrolyte volume fraction.

Combination Examples

5. Cell SOC vs. Electrode SOC

Key Distinction

Cell SOC ≠ a given electrode's SOC. During discharge, the two electrodes evolve in opposite directions:

SOC Relationship Governed by Lithium Conservation

Moles of lithium lost by the anode = Moles of lithium gained by the cathode:

Key insight: This relationship is exactly linear — it is a direct consequence of mass conservation, not an approximation.

N/P Capacity Ratio

Cell-Level SOC → Electrode SOC Automatic Conversion

In parameters.py, simply set battery_soc_initial = 0.3 (cell starts at 30%), and the program computes automatically.

graph TD
    A["battery_soc_initial = 0.3"] --> B["Endpoint Definition<br/>SOC_n_full=0.95, SOC_n_empty=0.05<br/>SOC_p_full=0.10"]
    B --> C["N/P Conservation<br/>SOC_p_empty = 0.10 + 0.90×(N/P)"]
    C --> D["Linear Interpolation<br/>SOC_n = 0.95 − 0.7×0.90 = 0.32<br/>SOC_p = 0.10 + 0.7×(N/P exchange rate)"]
    D --> E["Concentration Conversion<br/>c_init = SOC × c_max"]

6. OCV Curves (Open-Circuit Voltage)

OCV is the "fingerprint" of an electrode material — each material has a unique OCV–SOC relationship.

6.1 The Nernst Equation (Thermodynamic Foundation)

Physical Origin: The Nernst equation derives from the Gibbs free energy \(\Delta G = \Delta G_0 + RT \cdot \ln(\mathrm{activity\ ratio})\); it generates a complete curve with only 2 parameters. However, it assumes the material is an "ideal solution" — atoms do not interact with each other.

6.2 Ideal vs. Real

6.3 Nernst Equation Breakdown

The core of Nernst is simply \(\ln(\theta/(1-\theta))\), multiplied by the thermal voltage \(RT/F \approx 0.026\mathrm{V}\):

• \(\theta = 0.5\): \(\ln(1) = 0\), \(OCV = U_0\) (equilibrium)
• \(\theta \to 0\): \(\ln \to -\infty\), OCV drops steeply
• \(\theta \to 1\): \(\ln \to +\infty\), OCV rises steeply

Key Insight: The "logarithmic scale" is compressed into the "millivolt scale" by the thermal voltage — the shape of the curve is entirely a mathematical expression of the entropy of mixing.

6.4 Graphite Anode OCV (ocv_negative)

def ocv_negative(theta):
    U = (0.6379                                          # baseline
         + 0.5416 * np.exp(-305.5309 * theta)             # exponential spike (θ<3%)
         - 0.0440 * np.tanh((theta - 0.1958) / 0.1088)    # tanh step 1 (~20%)
         - 0.1978 * np.tanh((theta - 1.0571) / 0.0854)    # tanh step 2 (~100% cliff)
         - 0.6875 * np.tanh((theta + 0.0117) / 0.0529)    # tanh step 3 (main step!)
         - 0.0175 * np.tanh((theta - 0.5692) / 0.0875))   # tanh step 4 (minor correction)
    return U

Essentially, mathematical "building blocks" (tanh smooth steps + exp spike) are used to approximate the staircase-like OCV produced by graphite phase transitions:

Each \(\tanh\) step corresponds to one phase-transition stage of graphite during lithiation/delithiation.

6.5 NMC Cathode OCV (ocv_positive)

def ocv_positive(theta):
    return 4.15 - 0.9 * theta + 0.2 * theta**2

6.6 The Full Picture: Four-Term Superposition

Focus on the upper-right subplot — this is the OCV trajectory when discharging from 30% SOC to 0%:

From ~3.65 V (start) down to ~3.45 V (end).

Note: This is only the OCV contribution; the overpotential \(\eta\) has not been included yet (adding \(\eta\) will depress the voltage further).

7. The Four OCV Functions at a Glance

8. The SPM Four-Equation Panorama

graph TD
    P["parameters.py<br/>Battery physical parameters"] --> M["model.py"]

    subgraph model.py
        direction TB
        O["(1) OCV Curves<br/>U(θ): Electrode material fingerprint"]
        D["(2) Diffusion Equation<br/>∂c/∂t = D/r²·∂/∂r(r²·∂c/∂r)"]
        B["(3) Butler-Volmer<br/>j ⇄ η: Interfacial reaction kinetics"]
        V["(4) Voltage Coupling<br/>V_cell = U_p+η_p − U_n−η_n"]
        O --> B
        D --> V
        B --> V
    end

    M --> R["simulate.py<br/>Run & plot analysis"]

9. Project File Structure

c:\Project\P2D\
├── requirements.txt                # numpy, scipy, matplotlib
├── visualize_ocv.py                # OCV visualization script
├── learning_notes_01_foundation.md  # Study notes
├── ocv_comparison.png              # Figure 1: Nernst vs. empirical fit
├── nernst_breakdown.png            # Figure 2: Nernst equation breakdown
├── ocv_deep_dive.png               # Figure 3: Four-term superposition + discharge trajectory
├── electro-ref/                    # Reference books
└── spm/
    ├── __init__.py
    ├── parameters.py               # Battery physical parameters + automatic SOC conversion
    ├── model.py                    # SPM core engine (4 governing equations)
    └── simulate.py                 # To be created

10. Key Formula Quick Reference

11. To-Do / Next Steps

• Understand SPM model concepts
• Master parameter naming rules and physical meanings
• Understand N/P ratio and SOC conversion
• Understand OCV curves (Nernst + empirical fitting)
• Study the diffusion equation (Fick's Second Law in spherical coordinates)
• Study Butler-Volmer kinetics
• Study voltage coupling
• Write simulate.py, run a complete simulation
• Plot discharge curves, analyze results

End of Notes · Next Part: Diffusion Equation & Butler-Volmer Kinetics